Question

If f(x) = x²∛x, then f'(-8) = ?

Answer 1

To find the derivative of f(x) = x²∛x, apply the product and chain rules of differentiation. Substitute x = -8 to find f'(-8), which is undefined.

Step-by-step explanation:

To find the derivative of f(x) = x²∛x, we can apply the product and chain rules of differentiation:

1. Using the product rule, the derivative of x² is 2x, and the derivative of ∛x is ∛x/3.
2. Using the chain rule, we multiply the derivative of the outer function (x²) by the derivative of the inner function (∛x).
3. So, f'(x) = 2x*∛x/3.

To find f'(-8), we substitute x = -8 into f'(x):

1. f'(-8) = 2(-8)*∛(-8)/3 = -16*∛(-8)/3.
2. Since the cube root of a negative number is undefined, f'(-8) is also undefined.